Finite Group Databasebeta

← #10950930

Subgroups — #10950930

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#466C211yesmaximal, characteristic, complemented
2#83742C211yesmaximal, characteristic, complemented
3#72036C211yesmaximal, characteristic, complemented
4#72035C211yesmaximal, characteristic, complemented
5C2×#4549C211yesmaximal, characteristic, complemented
6#71076C212yesmaximal, complemented
7#71072C212yesmaximal, complemented
8#71071C212yesmaximal, complemented
9#71070C212yesmaximal, complemented
10#70603C211yesmaximal, characteristic, complemented
11#70600C211yesmaximal, characteristic, complemented
12#33754—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#121—21no—
16#5791—21no—
17C2×#904C2211yescharacteristic, complemented
18C3×#466—21no—
19C3×#466—21no—
20C3×#466C2212yes—
21C3×#466C2212yes—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#102—21no—
24C2×#827C2212yescomplemented
25C2×#826C2212yescomplemented
26#5116C2211yescharacteristic
27#5116C2211yescharacteristic, complemented
28#5115C2211yescharacteristic
29#5115C2211yescharacteristic, complemented
30#5114—21no—
31#5114—21no—
32#5113—21no—
33#5106C2211yescharacteristic, complemented
34C22×C3⋊C16C2212yescomplemented
35#4562C2212yescomplemented
36#4562C2212yescomplemented
37#4549C2211yescharacteristic
38#4549C2211yescharacteristic
39#4549C2212yescomplemented
40#4492C2212yescomplemented
41#4491C2212yescomplemented
42#4490C2212yescomplemented
43#4489C2212yescomplemented
44#3050—31no—
45C2×#466D311yescharacteristic, complemented
46#2236—31no—
47#2234—32no—
48#2233—31no—
49#2217—32no—
50#2215—32no—
51#2187—31no—
52#2186—32no—
53#2184—31no—
54C2×C22⋊C16—31no—
55C22×C3×D4—21no—
56C22×C3×D4—41no—
57C23×C3×C4D411yescharacteristic
58C2×#235—21no—
59C2×C3×D8—21no—
60C2×C3×D8—42no—
61C2×C3×D8C2312yes—
62C22×C3×C8—42no—
63C22×C3×C8D411yescharacteristic
64C22×C3×C8D411yescharacteristic
65C22×C3×C8D411yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C3×#121—22no—
70C3×#121—22no—
71C3×#121—41no—
72C3×#121—41no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C22⋊C4—41no—
75#953—41no—
76#930—21no—
77#930—21no—
78#929—41no—
79C4×#67—41no—
80C4×#67—41no—
81#926—21no—
82#924—21no—
83#924—21no—
84C2×Dic12C2311yescharacteristic
85#904—42no—
86C2×D24C2311yescharacteristic
87C2×#204—21no—
88C3×#107—21no—
89C3×#107C2312yes—
90C3×#102—22no—
91C3×#102—22no—
92#827—42no—
93#826—42no—
94#821—21no—
95#821—21no—
96#818C2311yescharacteristic
97#818C2311yescharacteristic
98#817—21no—
99#816—21no—
100#816—21no—
101C2×C3⋊C16—42no—
102C2×C3⋊C16C2312yes—
103C2×C3⋊C16C2312yes—
104C2×#135—31no—
105C22×D8C2×D311yescharacteristic
106C23×C8C2×D311yescharacteristic
107#525—61no—
108C2×#121—21no—
109C2×SD32—32no—
110C2×D16—32no—
111C22×C16—32no—
112#467—31no—
113#467—31no—
114#466—21no—
115#466—21no—
116#466—31no—
117#466—31no—
118#466C2×D312yes—
119#466C2×D312yes—
120#465—61no—
121#465—61no—
122#427—31no—
123C2×#107C2×D311yescharacteristic
124#416—61no—
125C2×#102—21no—
126#367—32no—
127#366—32no—
128#358—32no—
129#357—32no—
130#357—32no—
131#357—32no—
132C22⋊C16—31no—
133C22⋊C16—32no—
134C22⋊C16—31no—
135C24×C3—41no—
136C24×C3D811yescharacteristic
137C2×C3×D4—22no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—42no—
142C2×C3×D4—41no—
143C2×C3×D4—84no—
144C22×C3×C4—41no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4D811yescharacteristic
148C22×C3×C4D811yescharacteristic
149C22×C3×C4D811yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C2×#67—41no—
154C2×#67—41no—
155C2×#67—41no—
156#235—42no—
157#235—42no—
158C2×D12—21no—
159C2×C4×D3—41no—
160C2×Dic6—21no—
161C3×D8—22no—
162C3×D8—22no—
163C3×D8—42no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—41no—
168C2×C3×C8—41no—
169C2×C3×C8—41no—
170C2×C3×C8C2×D411yescharacteristic
171C2×C3×C8C2×D411yescharacteristic
172C2×C3×C8C2×D411yescharacteristic
173C2×C3×C8C2411yescharacteristic
174C3×C4⋊C4—22no—
175C3×C4⋊C4—41no—
176C3×C22⋊C4—42no—
177C3×C22⋊C4—42no—
178#217—41no—
179#217—41no—
180#217—41no—
181(C2×D3)⋊C4—41no—
182(C2×D3)⋊C4—41no—
183C4⋊Dic3—21no—
184C4⋊Dic3—21no—
185Dic3⋊C4—41no—
186Dic3⋊C4—41no—
187C4×Dic3—41no—
188Dic12—22no—
189D24—22no—
190#204—42no—
191C3⋊C16—22no—
192C3⋊C16—22no—
193C2×#41—61no—
194C22×D4—21no—
195C22×D4—41no—
196C23×C4#6711yescharacteristic
197#135—122no—
198C2×Q16—31no—
199C2×SD16—61no—
200C2×D8—21no—
201C2×D8—31no—
202C2×D8—42no—
203C2×D8C22×D312yes—
204C22×C8—42no—
205C22×C8#6711yescharacteristic
206C22×C8#6711yescharacteristic
207C22×C8#6711yescharacteristic
208C22×C8C22×D311yescharacteristic
209C22×C8C22×D311yescharacteristic
210C22×C8C22×D311yescharacteristic
211#123—121no—
212C22⋊Q8—61no—
213C22⋊Q8—61no—
214#121—22no—
215#121—22no—
216#121—41no—
217#121—41no—
218#121—61no—
219#121—61no—
220#120—121no—
221C4×D4—121no—
222C4×D4—121no—
223#117—61no—
224C2×C4⋊C4—21no—
225C2×C22⋊C4—41no—
226SD32—122no—
227D16—122no—
228C2×C16—32no—
229C2×C16—32no—
230C2×C16—122no—
231#107—21no—
232#107—31no—
233#107—31no—
234#107C22×D312yes—
235#106—61no—
236#103—61no—
237#103—61no—
238#102—22no—
239#102—22no—
240#102—61no—
241#102—61no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—41no—
245C23×C3—42no—
246C23×C3—82no—
247C23×C3—82no—
248C23×C3—82no—
249C23×C3C2×D811yescharacteristic
250C23×C3C2×D811yescharacteristic
251C23×C3C2×D811yescharacteristic
252C22×D3—41no—
253C3×D4—42no—
254C3×D4—42no—
255C3×D4—82no—
256C3×D4—84no—
257C3×D4—84no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—82no—
269C2×C3×C4C2×D811yescharacteristic
270C2×C3×C4C2×D811yescharacteristic
271C2×C3×C4C2×D811yescharacteristic
272C2×C3×C4C22×D411yescharacteristic
273#67—82no—
274#67—82no—
275#67—82no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3—41no—
279D12—41no—
280D12—41no—
281C4×D3—82no—
282Dic6—41no—
283Dic6—41no—
284C3×C8—21no—
285C3×C8—21no—
286C3×C8—21no—
287C3×C8H(5,𝔽2)12yes—
288C24—41no—
289C24#21311yescharacteristic
290#41—122no—
291#41—122no—
292C2×Q8—61no—
293C2×D4—22no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—61no—
300C2×D4—84no—
301C2×D4—121no—
302C2×D4—121no—
303C2×D4—121no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—121no—
308C22×C4#21311yescharacteristic
309C22×C4#21311yescharacteristic
310C22×C4#21311yescharacteristic
311C22×C4C2×#6711yescharacteristic
312C22×C4C2×#6711yescharacteristic
313C22×C4C2×#6711yescharacteristic
314Q16—62no—
315SD16—122no—
316D8—22no—
317D8—22no—
318D8—42no—
319D8—62no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8C2×#6711yescharacteristic
327C2×C8C2×#6711yescharacteristic
328C2×C8C2×#6711yescharacteristic
329C2×C8C23×D311yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—41no—
332C4⋊C4—61no—
333C4⋊C4—61no—
334C4⋊C4—121no—
335C4⋊C4—121no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—121no—
339C22⋊C4—121no—
340C22⋊C4—121no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C42—121no—
344C16—62no—
345C16—62no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—44no—
354C22×C3—82no—
355C22×C3—82no—
356C22×C3—82no—
357C22×C3—164no—
358C22×C3#50812yes—
359C22×C3#50912yes—
360C22×C3C22×D811yescharacteristic
361C2×D3—81no—
362C2×D3—81no—
363C2×D3—81no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—82no—
368C3×C4#53411yescharacteristic
369C3×C4#57511yescharacteristic
370Dic3—81no—
371Dic3—81no—
372Dic3—81no—
373C23—22no—
374C23—22no—
375C23—41no—
376C23—42no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—121no—
381C23C2×#21311yescharacteristic
382C23C2×#21311yescharacteristic
383C23C2×#21311yescharacteristic
384Q8—121no—
385Q8—121no—
386D4—42no—
387D4—42no—
388D4—82no—
389D4—84no—
390D4—84no—
391D4—121no—
392D4—121no—
393D4—242no—
394D4—242no—
395D4—242no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—242no—
411C2×C4C2×#21311yescharacteristic
412C2×C4C2×#21311yescharacteristic
413C2×C4C2×#21311yescharacteristic
414C2×C4C22×#6711yescharacteristic
415C8—21no—
416C8—21no—
417C8—21no—
418C8#100412yes—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—41no—
422C2×C3—162no—
423C2×C3#318611yescharacteristic
424C2×C3C2×#50811yescharacteristic
425C2×C3C2×#50911yescharacteristic
426D3—161no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—44no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22—164no—
439C22—241no—
440C22—241no—
441C22—241no—
442C22#514712yes—
443C22#519412yes—
444C22C22×#21311yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—82no—
449C4—241no—
450C4—241no—
451C4—241no—
452C4#580411yescharacteristic
453C4#582211yescharacteristic
454C3#3375411yescharacteristic, complemented, Sylow
455C2—22no—
456C2—22no—
457C2—41no—
458C2—162no—
459C2—481no—
460C2#8262511yescharacteristic
461C2C2×#514711yescharacteristic
462C2C2×#519411yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC2×C3×#466C2
socleC22×C3C22×D8
hypercenterC2×#466D3
hypocenterC3#33754
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C3×#466C2semidirect
2. #83742C2semidirect
3. #72036C2semidirect
4. #72035C2semidirect
5. C2×#4549C2semidirect
6. #71076C2semidirect
7. #71072C2semidirect
8. #71071C2semidirect
9. #71070C2semidirect
10. #70603C2semidirect
11. #70600C2semidirect
17. C2×#904C22semidirect
24. C2×#827C22semidirect
25. C2×#826C22semidirect
27. #5116C22semidirect
29. #5115C22semidirect
33. #5106C22semidirect
34. C22×C3⋊C16C22semidirect
35. #4562C22semidirect
36. #4562C22semidirect
39. #4549C22semidirect
40. #4492C22semidirect
41. #4491C22semidirect
42. #4490C22semidirect
43. #4489C22semidirect
45. C2×#466D3semidirect
454. C3#33754semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#4662
#837422
#720362
#720352
C2×#45492
#710762
#710722
#710712
#710702
#706032
#706002
#337543